On the characterization of a class of binary operations on distribution functions
type: Item
instance of: scholarly article
title: On the characterization of a class of binary operations on distribution functions
zbMATH Open document ID: 0798.60023
DOI: 10.1016/0167-7152(93)90001-Y
author: Claudi Alsina
author: Roger B. Nelsen
author: Berthold Schweizer
published in: Statistics \& Probability Letters
publication date: 1994-11-01T00:00:00Z
review text: The distribution function \(H\) of the sum of two (independent) random variables with distribution functions \(F\) and \(G\), respectively, is the convolution of \(F\) and \(G\). This function \(H\) can be viewed as the result of a binary operation on random variables or as the result of a binary operation on distribution functions. The authors completely characterize certain binary operations on distribution functions; namely, those that are induced pointwise by a function \(\Phi\) and can also be derived from a binary operation \(V\) on random variables. This collection of binary operations is surprisingly small. Indeed in Theorem 1 the authors show that \(\Phi\) must be a quasi copula and \(V\) must be Max, \(\Phi\) must be the dual of a quasi copula and \(V\) must be Min, or \(\Phi\) and \(V\) must both be trivial \((V(x,y)=y\), \(\Phi (a,b) = b\) or \(V(x,y) = x\), \(\Phi (a,b) = a)\). As a result, it follows that certain natural operations or distribution functions such as the mixture of two distributions can not be derived from a binary operation on random variables. Thus binary operations on random variables lead to a corresponding binary operation on distributions, but not conversely.
Mathematics Subject Classification ID: 60E99
Mathematics Subject Classification ID: 60A99
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 203386
zbMATH Keywords: copula
zbMATH Keywords: convolution
zbMATH Keywords: binary operations on distribution functions
zbMATH Keywords: quasi copula
zbMATH Keywords: mixture of two distributions
reviewed by: Robert M. Tardiff
MaRDI profile type: Publication
cites work: Operations on distribution functions not derivable from operations on random variables
cites work: Q3336431
full work available at URL: 0167-7152(93)90001-y
OpenAlex ID: W2057805381
description: scientific article; zbMATH DE number 203386